arxiv: 2605.10857 · v1 · submitted 2026-05-11 · 🌌 astro-ph.GA

Recognition: 2 theorem links

· Lean Theorem

Milky Way Dynamics Favor Dark Matter over Modified Gravity Models

Zheng-long Wang , Yue-Lin Sming Tsai , Lan Zhang , Yin Wu , Haining Li , Xiang-Xiang Xue , Hongsheng Zhao , Yi-Zhong Fan

Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:46 UTC · model grok-4.3

classification 🌌 astro-ph.GA

keywords dark mattermodified gravityMONDSTVGMilky WayGaiarotation curvegalactic dynamics

0 comments

The pith

Milky Way data shows modified gravity cannot fit both radial and vertical fields at once, while dark matter can.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper combines the Milky Way's radial rotation curve with vertical phase-space spirals from Gaia and a broken-exponential model of the stellar disk to test alternatives to dark matter. It reconstructs the gravitational field in both directions and finds that modified gravity models produce an internal contradiction: no single law can match the two independent constraints simultaneously. This rules out MOND at more than 13 sigma and STVG at more than 4 sigma. Dark matter halo models, by contrast, satisfy both constraints without contradiction. A sympathetic reader cares because the test breaks the usual degeneracy between unknown mass and unknown gravity on galactic scales.

Core claim

A joint reconstruction of the radial and vertical gravitational fields reveals a structural inconsistency in modified gravity -- no model can simultaneously reproduce both observations. Our results strongly disfavor MOND at >13σ and STVG at >4σ. In contrast, dark matter halo models naturally explain the observations, providing a self-consistent test of gravity on galactic scales.

What carries the argument

Joint reconstruction of radial and vertical gravitational fields from the rotation curve, Gaia vertical phase-space spirals, and broken-exponential stellar disk, which removes the usual degeneracy between baryonic structure and the form of gravity.

If this is right

Dark matter halo models remain viable while modified gravity models are excluded on galactic scales by this test.
Vertical kinematic data can be combined with radial data to test gravity independently of assumptions about the stellar mass distribution.
The same inconsistency appears for both MOND and STVG, suggesting the problem is generic to modified-gravity approaches that alter the force law without adding unseen mass.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same joint radial-vertical approach could be tried on external galaxies with good vertical kinematic data to check whether the result generalizes.
If the vertical-spiral constraint holds, it tightens the requirements any successful dark-matter model must satisfy for the Milky Way.
Future Gaia data releases with larger samples could raise or lower the significance levels reported here.

Load-bearing premise

The Gaia vertical phase-space spirals supply an independent measure of the vertical gravitational field that does not depend on the radial rotation curve or the precise shape assumed for the stellar disk.

What would settle it

Discovery of a single modified-gravity force law that reproduces both the measured radial rotation curve and the vertical gravitational field extracted from the Gaia spirals would refute the claimed inconsistency.

Figures

Figures reproduced from arXiv: 2605.10857 by Haining Li, Hongsheng Zhao, Lan Zhang, Xiang-Xiang Xue, Yin Wu, Yi-Zhong Fan, Yue-Lin Sming Tsai, Zheng-long Wang.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

read the original abstract

Modified gravity theories such as Modified Newtonian Dynamics (MOND) and Scalar-Tensor-Vector Gravity (STVG) have been proposed as alternatives to dark matter, but decisive tests have been hindered by degeneracies between baryonic structure and gravitational laws. Here we break this degeneracy using independent, high-precision constraints: the Milky Way radial rotation curve, vertical phase-space spirals from Gaia, and a broken-exponential stellar disk. A joint reconstruction of the radial and vertical gravitational fields reveals a structural inconsistency in modified gravity -- no model can simultaneously reproduce both observations. Our results strongly disfavor MOND at $>13\sigma$ and STVG at $>4\sigma$. In contrast, dark matter halo models naturally explain the observations, providing a self-consistent test of gravity on galactic scales.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper combines Gaia vertical phase-space spirals with the Milky Way rotation curve to claim a structural mismatch that rules out MOND at >13σ and STVG at >4σ, but the vertical constraint likely shares dependence on the broken-exponential disk model, weakening the independence.

read the letter

The main thing to know is that this work uses two Milky Way dynamical probes to test modified gravity directly. They reconstruct radial and vertical fields and find no single MOND or STVG model fits both, while a dark matter halo does. The Gaia spirals supply the vertical piece and the rotation curve the radial one, with a broken-exponential disk for the baryons. That combination is the new element here, and it is a reasonable way to attack the usual degeneracy between baryonic structure and the gravity law. The paper does a clean job of laying out why dark matter accommodates the data without extra tuning. The high significance numbers against the alternatives are presented as coming from the mismatch itself rather than from priors alone. That part is worth looking at closely. The soft spot is the claimed independence of the two constraints. Deriving the vertical force from the phase-space spirals requires a model for the stellar density and vertical structure, and the authors use the same broken-exponential profile that enters the radial fit. In modified gravity the total field is fixed by the baryons, so modest shifts in disk scale length, mass, or break radius within current uncertainties can move the inferred vertical field enough to reduce or remove the tension. The abstract treats the inconsistency as structural and data-driven, but the joint reconstruction introduces some shared modeling that is not fully decoupled. Without the full error budget and marginalization details it is hard to judge whether the >13σ and >4σ figures survive that variation. This paper is for people who follow galactic dynamics tests and the dark matter versus modified gravity debate. A reader who already works with Gaia kinematics or rotation curve modeling will get the most out of the figures and the specific parameter choices. It deserves a serious referee because the data combination is new and the question is well-posed, even though the independence claim needs tighter checking on the disk assumptions.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that a joint reconstruction of the Milky Way's radial and vertical gravitational fields—using the observed radial rotation curve, vertical phase-space spirals from Gaia, and a broken-exponential stellar disk model—reveals a structural inconsistency in modified gravity. No MOND or STVG model can simultaneously reproduce both fields, leading to disfavoring of MOND at >13σ and STVG at >4σ, while dark matter halo models provide a self-consistent explanation.

Significance. If the claimed independence of the vertical constraint holds and the error budgets are robust, this would constitute a significant test of gravity on galactic scales by breaking the baryon-gravity degeneracy that has limited prior comparisons. The incorporation of high-precision Gaia phase-space data and the quantitative tension levels reported would strengthen the empirical case for dark matter over modified gravity alternatives if the analysis withstands scrutiny of parameter dependencies.

major comments (2)

[vertical phase-space spirals analysis and joint reconstruction] The central claim requires that the Gaia vertical phase-space spirals yield a gravitational field measurement independent of the assumed broken-exponential stellar disk profile and the radial rotation curve. However, deriving the vertical force from phase-space spirals involves modeling the underlying stellar density and vertical structure with the same broken-exponential profile used for the radial fit. In modified gravity, where the total field is directly determined by the baryonic distribution (unlike a separate DM halo), even modest variations in disk scale length, mass, or break radius within current uncertainties can shift the inferred vertical field enough to allow a joint fit, lowering the reported >13σ (MOND) and >4σ (STVG) tensions. This assumption is load-bearing for the exclusion significance and requires explicit tests (e.g., marginalization over disk parameters or Monte-C,
[Abstract and Methods] The abstract states high-significance results from a joint field reconstruction, but without the full methods, data selection criteria, error budgets, or model parameterizations (including how the broken-exponential disk parameters and MOND/STVG acceleration scales are jointly constrained), it is impossible to verify whether post-hoc choices or unaccounted systematics affect the claimed sigma levels. The reconstruction necessarily involves fitting disk and halo parameters to the same data sets used for the test, which raises the need for a clear demonstration that the two observational constraints remain independent after accounting for shared modeling assumptions.

minor comments (1)

Clarify in the text and figures how the vertical field is extracted from the phase-space spirals (e.g., which moments or distribution functions are used) and ensure all model curves are overlaid with the same disk parameter choices for direct comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify key aspects of our analysis. We address the major comments point by point below and will revise the manuscript to incorporate additional tests and expanded methodological details.

read point-by-point responses

Referee: [vertical phase-space spirals analysis and joint reconstruction] The central claim requires that the Gaia vertical phase-space spirals yield a gravitational field measurement independent of the assumed broken-exponential stellar disk profile and the radial rotation curve. However, deriving the vertical force from phase-space spirals involves modeling the underlying stellar density and vertical structure with the same broken-exponential profile used for the radial fit. In modified gravity, where the total field is directly determined by the baryonic distribution (unlike a separate DM halo), even modest variations in disk scale length, mass, or break radius within current uncertainties can shift the inferred vertical field enough to allow a joint fit, lowering the reported >13σ (MOND) and >4σ (STVG) tensions. This assumption is load-bearing for the exclusion significance and requires expl

Authors: We agree that the shared broken-exponential disk model warrants explicit validation to confirm robustness. The radial rotation curve constrains the azimuthally averaged mass distribution across a wide radial range, while the Gaia vertical phase-space spirals provide an independent local probe of the vertical gravitational acceleration in the solar neighborhood, derived from the vertical motions of stars. In modified gravity the total field must be sourced solely by baryons, so any viable model must satisfy both datasets with the same disk parameters. Nevertheless, to directly address the concern, we will add Monte Carlo marginalization over disk parameters (scale length, mass, break radius) within their observational uncertainties and show that the tensions persist at high significance. These tests and updated figures will be included in the revised Methods and Supplementary Material. revision: yes
Referee: [Abstract and Methods] The abstract states high-significance results from a joint field reconstruction, but without the full methods, data selection criteria, error budgets, or model parameterizations (including how the broken-exponential disk parameters and MOND/STVG acceleration scales are jointly constrained), it is impossible to verify whether post-hoc choices or unaccounted systematics affect the claimed sigma levels. The reconstruction necessarily involves fitting disk and halo parameters to the same data sets used for the test, which raises the need for a clear demonstration that the two observational constraints remain independent after accounting for shared modeling assumptions.

Authors: We acknowledge that the current presentation would benefit from greater methodological transparency. In the revised manuscript we will substantially expand the Methods section to provide: (i) explicit data selection and quality cuts for both the rotation curve and Gaia phase-space spirals, (ii) a complete error budget separating statistical, systematic, and modeling uncertainties, (iii) full parameterization details for the broken-exponential disk and the MOND/STVG acceleration scales together with the joint-fitting procedure, and (iv) a dedicated subsection with cross-validation tests demonstrating that the radial and vertical constraints remain effectively independent once shared parameters are accounted for. These additions will allow readers to reproduce and verify the reported significance levels. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's derivation uses the radial rotation curve and Gaia vertical phase-space spirals as distinct observational inputs to reconstruct gravitational fields, with the broken-exponential disk serving as a fixed baryonic model rather than a derived output. The claimed structural inconsistency for modified gravity arises from the inability of any single MG law to match both fields simultaneously, while DM halos accommodate them. No step reduces by construction to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation; the vertical constraint is explicitly presented as independent of the disk profile and radial curve degeneracies. The overall chain remains self-contained against external data benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the full set of modeling choices cannot be audited. The analysis rests on standard galactic dynamics assumptions plus specific choices for the stellar disk profile and field reconstruction whose details are not extractable here.

free parameters (2)

broken-exponential disk parameters
Scale lengths, densities, and break radii for the stellar disk model used to compute baryonic contribution.
MOND and STVG acceleration scales
Theory-specific parameters that may be fixed or adjusted during the fit.

axioms (1)

domain assumption Vertical phase-space spirals trace the vertical gravitational potential independently of radial dynamics.
Invoked to allow separate reconstruction of vertical field from Gaia data.

pith-pipeline@v0.9.0 · 5446 in / 1341 out tokens · 75118 ms · 2026-05-12T03:46:02.211817+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

A joint reconstruction of the radial and vertical gravitational fields reveals a structural inconsistency in modified gravity -- no model can simultaneously reproduce both observations.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

72 extracted references · 72 canonical work pages · 2 internal anchors

[1]

J. S. Bullock and M. Boylan-Kolchin, Annual Review of Astronomy and Astrophysics55, 343–387 (2017)

work page 2017
[2]

Banik and H

I. Banik and H. Zhao, Symmetry14, 1331 (2022)

work page 2022
[3]

Milgrom, The Astrophysical Journal270, 365 (1983)

M. Milgrom, The Astrophysical Journal270, 365 (1983)

work page 1983
[4]

Modiﬁed Newtonian Dynamics ( MOND): Observational Phenomenology and Relativistic Extensions

B. Famaey and S. S. McGaugh, Living Reviews in Rela- tivity 15, 10 (2012), arXiv:1112.3960 [astro-ph.CO]

work page arXiv 2012
[5]

S. S. McGaugh, F. Lelli, and J. M. Schombert, Physi- cal Review Letters117, 201101 (2016), arXiv:1609.05917 [astro-ph.GA]

work page arXiv 2016
[6]

McGaugh, Galaxies8, 35 (2020)

S. McGaugh, Galaxies8, 35 (2020)

work page 2020
[7]

Roshan, I

M. Roshan, I. Banik, N. Ghafourian, I. Thies, B. Famaey, E. Asencio, and P. Kroupa, Monthly Notices of the Royal Astronomical Society503, 2833–2860 (2021)

work page 2021
[8]

J. W. Moffat and S. Rahvar, Monthly Notices of the Royal Astronomical Society436, 1439 (2013)

work page 2013
[9]

Green and J

M. Green and J. Moffat, Physics of the Dark Universe 25, 100323 (2019)

work page 2019
[10]

Chae, Astrophys

K.-H. Chae, Astrophys. J. 952, 128 (2023), [Erratum: Astrophys.J. 956, 69 (2023)], arXiv:2305.04613 [astro- ph.GA]

work page arXiv 2023
[11]

K. H. Chae, B. C. Lee, X. Hernandez, V. G. Orlov, D. Lim, D. A. Turnshek, and Y. W. Lee, Detection of gravitational anomaly at low acceleration from a highest- quality sample of 36 wide binaries with accurate 3d ve- locities (2026), arXiv:2601.21728 [astro-ph.GA]

work page arXiv 2026
[12]

Haslbauer, I

M. Haslbauer, I. Banik, and P. Kroupa, Monthly No- tices of the Royal Astronomical Society499, 2845 (2020), arXiv:2009.11292 [astro-ph.CO]

work page arXiv 2020
[13]

G.W.Angus,MonthlyNoticesoftheRoyalAstronomical Society 394, 527–532 (2009)

work page 2009
[14]

Bílek and H

M. Bílek and H. Zhao, Astronomy & Astrophysics698, 10.1051/0004-6361/202452250 (2025)

work page doi:10.1051/0004-6361/202452250 2025
[15]

Russell, I

A. Russell, I. Banik, O. Cray, and H. Zhao, Monthly Notices of the Royal Astronomical Society 10.1093/mn- ras/stag399 (2026)

work page doi:10.1093/mn- 2026
[16]

M.Haslbauer, I.Banik, P.Kroupa, H.Zhao,andE.Asen- cio, Universe10, 385 (2024)

work page 2024
[17]

Banik, C

I. Banik, C. Pittordis, W. Sutherland, B. Famaey, R. Ibata, S. Mieske, and H. Zhao, Mon. Not. Roy. Astron. Soc. 527, 4573 (2024), arXiv:2311.03436 [astro-ph.SR]

work page arXiv 2024
[18]

S. Alam, M. Ata, S. Bailey,et al., Monthly Notices of the Royal Astronomical Society470, 2617 (2017)

work page 2017
[19]

Aghanim, Y

N. Aghanim, Y. Akrami, M. Ashdown,et al., Astronomy & Astrophysics641, A6 (2020)

work page 2020
[20]

Jurićet al., The Astrophysical Journal 673, 864 (2008)

M. Jurićet al., The Astrophysical Journal 673, 864 (2008)

work page 2008
[21]

Nipoti, P

C. Nipoti, P. Londrillo, H. S. Zhao, and L. Ciotti, Monthly Notices of the Royal Astronomical Society379, 597 (2007)

work page 2007
[22]

Lisanti, M

M. Lisanti, M. Moschella, N. J. Outmezguine, and O. Slone, Physical Review D 100, 083009 (2019), arXiv:1812.08169 [astro-ph.GA]

work page arXiv 2019
[23]

Davari and S

Z. Davari and S. Rahvar, Monthly Notices of the Royal Astronomical Society496, 3502 (2020)

work page 2020
[24]

Zhu, H.-X

Y. Zhu, H.-X. Ma, X.-B. Dong, Y. Huang, T. Mistele, B. Peng, Q. Long, T. Wang, L. Chang, and X. Jin, Monthly Notices of the Royal Astronomical Society519, 4479 (2022)

work page 2022
[25]

López-Corredoira, The Astrophysical Journal978, 45 (2025), arXiv:2412.09665 [astro-ph.GA]

M. López-Corredoira, The Astrophysical Journal978, 45 (2025), arXiv:2412.09665 [astro-ph.GA]

work page arXiv 2025
[26]

Antoja, A

T. Antoja, A. Helmi, M. Romero-Gómez, D. Katz, C. Babusiaux, R. Drimmel, D. W. Evans, F. Figueras, E. Poggio, C. Reylé, A. C. Robin, G. Seabroke, and C. Soubiran, Nature561, 360 (2018)

work page 2018
[27]

Guo, Z.-Y

R. Guo, Z.-Y. Li, J. Shen, S. Mao, and C. Liu, The As- trophysical Journal 960, 133 (2024), arXiv:2310.10225 [astro-ph.GA]

work page arXiv 2024
[28]

Bovy, H.-W

J. Bovy, H.-W. Rix, E. F. Schlafly, D. L. Nidever, J. A. Holtzman, M. Shetrone, and T. C. Beers, The Astrophys- ical Journal823, 30 (2016)

work page 2016
[29]

Ted Mackereth, J

J. Ted Mackereth, J. Bovy, R. P. Schiavon, G. Zasowski, K. Cunha, P. M. Frinchaboy, A. E. García Perez, M. R. Hayden, J. Holtzman, S. R. Majewski, S. Mészáros, D. L. Nidever, M. Pinsonneault, and M. D. Shetrone, Monthly Notices of the Royal Astronomical Society 471, 3057 (2017)

work page 2017
[30]

J. Lian, G. Zasowski, T. Mackereth, J. Imig, J. A. Holtzman, R. L. Beaton, J. C. Bird, K. Cunha, J. G. Fernández-Trincado, D. Horta, R. R. Lane, K. L. Mas- ters, C. Nitschelm, and A. Roman-Lopes, Monthly No- tices of the Royal Astronomical Society513, 4130 (2022)

work page 2022
[31]

Cantat-Gaudin, M

T. Cantat-Gaudin, M. Fouesneau, H.-W. Rix, A. G. A. Brown, R. Drimmel, A. Castro-Ginard, S. Khanna, V. Belokurov, and A. R. Casey, Astronomy & As- trophysics 683, A128 (2024), arXiv:2401.05023 [astro- ph.GA]

work page arXiv 2024
[32]

J. Imig, J. A. Holtzman, G. Zasowski, J. Lian, N. F. Boardman, A. Stone-Martinez, J. T. Mackereth, M.K.M.Prescott, R.L.Beaton, T.C.Beers, D.Bizyaev, M. R. Blanton, K. Cunha, J. G. Fernández-Trincado, C. E. Fielder, S. Hasselquist, C. R. Hayes, M. Haywood, H. Jönsson, R. R. Lane, S. R. Majewski, S. Mészáros, I. Minchev, D. L. Nidever, C. Nitschelm, and J. ...

work page 2025
[33]

Z. Yu, J. Li, B. Chen, Y. Huang, S. Jia, M. Xiang, H. Yuan, J. Shi, C. Wang, and X. Liu, The Astrophysical Journal 912, 106 (2021)

work page 2021
[34]

Z. Yu, B. Chen, J. Lian, C. Wang, and X. Liu, The Astro- nomical Journal169, 61 (2025), arXiv:2412.14743 [astro- ph.GA]

work page arXiv 2025
[35]

J. Lian, G. Zasowski, B. Chen, J. Imig, T. Wang, N. Boardman, and X. Liu, Nature Astronomy8, 1302 (2024), arXiv:2406.05604 [astro-ph.GA]

work page arXiv 2024
[36]

J. Lian, T. Wang, Q. Feng, Y. Huang, and H. Guo, The Astrophysical Journal Letter 990, L37 (2025), arXiv:2508.13665 [astro-ph.GA]

work page arXiv 2025
[37]

X.Ou, A.-C.Eilers, L.Necib,andA.Frebel,MonthlyNo- tices of the Royal Astronomical Society528, 693 (2024), arXiv:2303.12838 [astro-ph.GA]

work page arXiv 2024
[38]

S. R. Majewskiet al., Astron. J.154, 94 (2017)

work page 2017
[39]

Vallenariet al., Astron

A. Vallenariet al., Astron. Astrophys.674, A1 (2023)

work page 2023
[40]

M. F. Skrutskieet al., Astron. J.131, 1163 (2006)

work page 2006
[41]

E. L. Wrightet al., Astron. J.140, 1868 (2010)

work page 2010
[42]

Eilers, D

A.-C. Eilers, D. W. Hogg, H.-W. Rix, and M. K. Ness, The Astrophysical Journal 871, 120 (2019), arXiv:1810.09466 [astro-ph.GA]

work page arXiv 2019
[43]

Y. Zhou, X. Li, Y. Huang, and H. Zhang, The Astrophys- ical Journal946, 73 (2023)

work page 2023
[44]

J. Buch, J. S. C. Leung, and J. Fan, Journal of Cosmology and Astroparticle Physics 2019 (4), 026, arXiv:1808.05603 [astro-ph.GA]. 7

work page Pith review arXiv 2019
[45]

Cheng, B

X. Cheng, B. Anguiano, S. R. Majewski, and P. Arras, Monthly Notices of the Royal Astronomical Society527, 959 (2024), arXiv:2309.17405 [astro-ph.GA]

work page arXiv 2024
[46]

Söding, R

L. Söding, R. L. Bartel, and P. Mertsch, Monthly No- tices of the Royal Astronomical Society542, 2987 (2025), arXiv:2506.02956 [astro-ph.GA]

work page arXiv 2025
[47]

X. Ou, L. Necib, A. Wetzel, A. Frebel, J. Bailin, and M. Oeur, The Astrophysical Journal 994, 128 (2025), arXiv:2503.05877 [astro-ph.GA]

work page arXiv 2025
[48]

P. J. McMillan, Monthly Notices of the Royal Astronom- ical Society465, 76 (2016)

work page 2016
[49]

M. C. Sormani, O. Gerhard, M. Portail, E. Vasiliev, and J. Clarke, Monthly Notices of the Royal Astronomical Society: Letters514, L1 (2022)

work page 2022
[50]

Analytical Models For Galactic Nuclei

H. Zhao, Monthly Notices of the Royal Astronomical So- ciety278,488(1996),arXiv:astro-ph/9509122[astro-ph]

work page Pith review arXiv 1996
[51]

J. F. Navarro, C. S. Frenk, and S. D. M. White, The Astrophysical Journal490, 493 (1997), arXiv:astro- ph/9611107 [astro-ph]

work page arXiv 1997
[52]

Einasto, Trudy Astrofizicheskogo Instituta Alma-Ata 5, 87 (1965)

J. Einasto, Trudy Astrofizicheskogo Instituta Alma-Ata 5, 87 (1965)

work page 1965
[53]

J. W. Moffat, Journal of Cosmology and Astroparticle Physics 2006 (3), 004, arXiv:gr-qc/0506021 [gr-qc]

work page arXiv 2006
[54]

A. A. Dutton and A. V. Macciò, Monthly Notices of the Royal Astronomical Society441, 3359 (2014)

work page 2014
[55]

A. V. Macciò, A. A. Dutton, and F. C. van den Bosch, Monthly Notices of the Royal Astronomical Society391, 1940 (2008)

work page 1940
[56]

Di Cintio, C

A. Di Cintio, C. B. Brook, A. A. Dutton, A. V. Macciò, G. S. Stinson, and A. Knebe, Monthly Notices of the Royal Astronomical Society441, 2986 (2014)

work page 2014
[57]

Freundlich, A

J. Freundlich, A. Dekel, F. Jiang, G. Ishai, N. Cornuault, S.Lapiner, A.A.Dutton,andA.V.Macciò,MonthlyNo- tices of the Royal Astronomical Society491, 1190 (2020)

work page 2020
[58]

Milgrom and R

M. Milgrom and R. H. Sanders, The Astrophysical Jour- nal 678, 131–143 (2008)

work page 2008
[59]

V. Z. Adibekyan, S. G. Sousa, N. C. Santos, E. Delgado Mena, J. I. González Hernández, G. Israelian, M. Mayor, and G. Khachatryan, Astronomy & Astrophysics545, A32 (2012), arXiv:1207.2388 [astro-ph.EP]

work page arXiv 2012
[60]

Binney and S

J. Binney and S. Tremaine,Galactic Dynamics, 2nd ed. (PrincetonUniversityPress,2008)princetonSeriesinAs- trophysics

work page 2008
[61]

Aigrain and D

S. Aigrain and D. Foreman-Mackey, Annual Re- view of Astronomy and Astrophysics 61, 329 (2023), arXiv:2209.08940 [astro-ph.IM]

work page arXiv 2023
[62]

R. Guo, C. Liu, S. Mao, X.-X. Xue, R. J. Long, and L. Zhang, Monthly Notices of the Royal Astronomical So- ciety 495, 4828 (2020), arXiv:2005.12018 [astro-ph.GA]

work page arXiv 2020
[63]

2021, AJ, 161, 147, doi: 10.3847/1538-3881/abd806

C.A.L.Bailer-Jones, J.Rybizki, M.Fouesneau, M.Dem- leitner, and R. Andrae, The Astronomical Journal161, 147 (2021), arXiv:2012.05220 [astro-ph.SR]

work page internal anchor Pith review arXiv 2021
[64]

Milgrom, Monthly Notices of the Royal Astronomical Society 403, 886 (2010), arXiv:0911.5464 [astro-ph.CO]

M. Milgrom, Monthly Notices of the Royal Astronomical Society 403, 886 (2010), arXiv:0911.5464 [astro-ph.CO]

work page arXiv 2010
[65]

Zhao, The Astrophysical Journal671, L1–L4 (2007)

H. Zhao, The Astrophysical Journal671, L1–L4 (2007)

work page 2007
[66]

C. F. McKee, A. Parravano, and D. J. Hollenbach, The Astrophysical Journal814, 13 (2015)

work page 2015
[67]

Bayes in the sky: Bayesian inference and model selection in cosmology

R. Trotta, Contemporary Physics 49, 71 (2008), arXiv:0803.4089 [astro-ph]

work page Pith review arXiv 2008
[68]

K. P. Burnham and D. R. Anderson,Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach (Springer-Verlag, 2002)

work page 2002
[69]

Gelman, J

A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin,Bayesian Data Analysis, 3rd ed. (CRC Press, 2013)

work page 2013
[70]

emcee: The MCMC Hammer

D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Good- man, Publications of the Astronomical Society of the Pa- cific 125, 306 (2013), arXiv:1202.3665 [astro-ph.IM]

work page internal anchor Pith review arXiv 2013
[71]

Schwarz, Annals of Statistics6, 461 (1978)

G. Schwarz, Annals of Statistics6, 461 (1978)

work page 1978
[72]

Milky Way Dynamics Favor Dark Matter over Modified Gravity Models

Y. Huang, X.-W. Liu, H.-B. Yuan, M.-S. Xiang, H.-W. Zhang, B.-Q. Chen, J.-J. Ren, C. Wang, Y. Zhang, Y.- H. Hou, Y.-F. Wang, and Z.-H. Cao, Monthly Notices of the Royal Astronomical Society 463, 2623 (2016), arXiv:1604.01216 [astro-ph.GA]. S1 Supplemental Material for “Milky Way Dynamics Favor Dark Matter over Modified Gravity Models” S1. MATERIALS AND ME...

work page arXiv 2016